How to control the timing of Absolute Shape Keys?
Controlling the timing of absolute shape keys

You can add keyframes for the "Evaluation Time". The easiest way in the UI is to hover over the Evaluation Time slider and press "i". This will map your defined shape keys to particular frames in your animation.

Attempted answers to your other questions


The "frame" for absolute keys does seem to be separate from the usual meaning of "frame" in Blender and not related to keyframes. Not sure why it increments by 10 or why it's not editable, but it ends up not mattering much because of the fact that you can set the eval time and make keyframes for it.
Resetting timing seems to be an organizational tool. If you delete one of the shape keys, you'll have a gap of 20 in the eval time from one to the next, and resetting lets you get back to a list that increments by 10. If there's a deeper reason, I don't know it.
As far as I can tell, the best way to loop is, as you suggest, duplicating the basis keyframe for the final keyframe. I tried setting a final keyframe with evaluation time set to zero, and the result was my object rapidly undoing all of its shape changes, one at a time, on its way back to the basis shape key. To connect the loop after making a keyframe for your final shape key, I'd try making one more keyframe for your basis shape key in the very next frame to stop the intermediate values of eval time from getting animated. (If you want to avoid the doubling of a frame, you could also set eval time to a fractional value corresponding to what the shape of the object would be at that frame if it were continuously morphing.)